Counting bridgeless graphs

Methods are developed for finding the number of unlabeled bridgeless or 2-line-connected graphs of any order. These methods are based on cycle index sums, but it is shown how to avoid explicit compution with cycle index sums by using suitable inversion techniques. Similar results are obtained for unlabeled bridgeless graphs by numbers of points and lines, and connected graphs by numbers of points and bridges. Corresponding results for labeled graphs are found as corollaries. When lines or bridges are required as enumeration parameters in the labeled case it is also shown how to obtain improved recurrence relations. The latter appear to have no analog for unlabeled graphs.